Question: Find the smallest positive integer $b$ for which $x^2 + bx + 2008$ factors into a product of two polynomials, each having integer coefficients.
Explanation: We can let the factorization be
\[x^2 + bx + 2008 = (x + p)(x + q),\]where $p$ and $q$ are integers.  Then $p + q = b$ and $pq = 2008.$

The equation $pq = 2008$ tells us that either both $p$ and $q$ are positive, or both are negative.  Since $p + q = b$ is positive, both $p$ and $q$ are positive.

We want to find the minimum value of $b.$  The number $b = p + q$ is minimized when $p$ and $q$ are as close as possible, under the condition $pq = 2008.$  This occurs when $p$ and $q$ are 8 and 251, so the smallest possible value of $b$ is $8 + 251 = \boxed{259}.$